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# Publications of the Research Institute for Mathematical Sciences

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**Volume 53, Issue 2, 2017, pp. 261–285**

**DOI: 10.4171/PRIMS/53-2-2**

Published online: 2017-05-04

$\Gamma$-Unitaries, Dilation and a Natural Example

Tirthankar Bhattacharyya^{[1]}and Haripada Sau

^{[2]}(1) Indian Institute of Science, Bangalore, India

(2) Indian Institute of Science, Bangalore, India

This note constructs an explicit normal boundary dilation for a commuting pair $(S,P)$ of bounded operators with the symmetrized bidisk $$\Gamma=\{(z_1+z_2,z_1z_2):|z_1|,|z_2| \leq 1\}$$ as a spectral set. Such explicit dilations had hitherto been constructed only in the unit disk [11], the unit bidisk [3] and in the tetrablock [6]. The dilation is minimal and unique under a suitable condition. This paper also contains a natural example of a $\Gamma$-isometry. We compute its associated fundamental operator.

*Keywords: *Symmetrized bidisk, spectral set, $\Gamma$-contraction, $\Gamma$-unitary, dilation

Bhattacharyya Tirthankar, Sau Haripada: $\Gamma$-Unitaries, Dilation and a Natural Example. *Publ. Res. Inst. Math. Sci.* 53 (2017), 261-285. doi: 10.4171/PRIMS/53-2-2